If the average speed of an orbiting space shuttle is 19600 mi/h, determine the time required for it to circle the Earth. Make sure you consider the fact that the shuttle is orbiting about 200 mi above the Earth's surface, and assume that the Earth's radius is 3963 miles

However, the velocity of a satellite orbiting at 200 miles altitude is only 17,256mph.

If your intent was that the orbit is elliptical, you would need the perigee and apogee velocities to average 19,600mph. Since the perigee velocity is already only 17,256mph, and the apogee velocity is less, an average of 19,600 is impossible.

If the average velocity of 19,600 mph is meant to be the average of the perigee and apogee velocities of an elliptical orbit, you would have to derive an elliptical orbit having a perigee velocity higher than 17,256 mph and an apogee velocity that averages 19,600mph when added to the perigee velocity.

The perigee velocity is defined by
Vp = [2µ(Ra/(Ra+Rp)]^1/2 while the apogee velocity is defined by
Va = [2µ(Rp/(Rp+Ra)]^1/2.

You will have to play with these expressions until you find an elliptical orbit whose perigee and apogee velocities average 19,600 mph.

If the average velocity of 19,600 mph is meant to be the average of the perigee and apogee velocities of an elliptical orbit, you would have to derive an elliptical orbit having a perigee velocity higher than 17,256 mph and an apogee velocity that averages 19,600mph when added to the perigee velocity.

The perigee velocity is defined by
Vp = [2µ(Ra/(Ra+Rp)]^1/2 while the apogee velocity is defined by
Va = [2µ(Rp/(Rp+Ra)]^1/2 where Ra and Rp = the apogee and perigee radii respectively.

You will have to play with these expressions until you find an elliptical orbit whose perigee and apogee velocities average 19,600 mph.

The perigee velocity must be lower than 24,403 mph or the satellite will escape earth's gravity, never to return.

Considering that the perigee velocity must be less than 24,403 mph, if it were 24,402 and the companion apogee velocity very small, say 1 mph, the average of the two would be far less than 19,600 mph, indicating that it is impossible.